The Prince of Wales was one of the first to congratulate the Welsh team on their 32-20 victory over the Irish.
He said: "I knew you would do it. One thousand congratulations and many admiring wishes on such a spectacular Welsh triumph - llongyfarchiadau (Welsh for congratulations)."
Welsh secretary Peter Hain heaped praise on the team's efforts.
He said: "What a fantastic performance by Wales. Winning all our Six Nations games - with three of them away from home - is a magnificent achievement.
"This is one of Wales' great sporting moments when people will be proud to say in years to come, 'I was there'."
A coin commemorating the Grand Slam season will go into production at the Royal Mint in Llantrisant.
Police say that the celebrations after the historic win also passed off relatively peacefully despite 200,000 fans descending on Cardiff for the game.
Chief Superintendent Bob Evans, of South Wales Police, said: "It was an incredible day for the city with an unprecedented number of visitors in and around the stadium and city centre.
"We think up to 200,000 people over and above a normal Saturday were in the city with 15-20,000 who watched the match on the big screen in Cathays Park.
"There was a great atmosphere and we only had a couple of casualties during the match period with a few people being treated for having too much to drink.
"Overall there were 42 arrests in the city but mostly for incidents of minor public order and not directly linked to the final."
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